Why some constraints work in view port, but don't work while render? (Blender 2.8)
No, it's not a bug.

This is actually known issue called "dependency cycle"

If you open blender with terminal, you will see like this:

Dependency cycle detected:
  'OBCamera.TRANSFORM_CONSTRAINTS()' depends on 'OBEmpty.TRANSFORM_FINAL()' through 'Damped Track'
  'OBEmpty.TRANSFORM_FINAL()' depends on 'OBEmpty.TRANSFORM_CONSTRAINTS()' through 'ObConstraints -&gt; Done'


This means that one constraint depends on second, second on first one. So current camera rotation cannot be calculated correctly, because there is no start point to start with.

To solve this, fix dependency in your scene


  I've tried "blender --enable-new-depsgraph" with blender 2.79b and it
  still works despite the new kind of the error messages


You will get same errors in 2.79:

Dependency cycle detected:
  Empty depends on Camera through Copy Location.
  Camera depends on Empty through Damped Track.


And the fact that 2.79 in this error situation doing something different, does not mean that 2.80 has a bug or something. Look at glitch with camera movement when I change frame:

  

The only way to solve this problem - do not build cycles in dependencies. Add an empty for follow path, second empty will follows this one, and camera will depends on first and second empties:


Camera moved a little for clarity

No dependency cycles - no problems.